Q.Bag I contains $4$ white and $3$ black balls, and Bag II contains $3$ white and $5$ black balls. A bag is chosen at random and a ball is drawn from it. Find the probability that the ball drawn is white.

[!TLDR] Total probability: $P(W)=P(\text{I})P(W\mid\text{I})+P(\text{II})P(W\mid\text{II …

Each bag is equally likely to be chosen, so $P(\text{I})=P(\text{II})=\tfrac12$. From Bag I, $P(W\mid\text{I})=\tfrac47$; from Bag II, $P(W\mid\text{II})=\tfrac38$. By the total probability theorem, $P(W)=\tfrac12\times\tfrac47+\tfrac12\times\tfrac38=\tfrac{4}{14}+\tfrac{3}{16}$. Over a common denominator of $112$: $\tfrac{4}{14}=\ …

Identify the partition (which bag is chosen), find $P(W\mid\text{bag})$ for each bag, then com …

Averaging the two conditional probabilities $\tfrac47$ and $\tfrac38$ directly (adding fractions without a common denominator, or forgetting …